Symmetric A actions on $\mathcal{A}(2)$

We describe the variety of `symmetric' left actions of the mod 2 Steenrod algebra $\mathcal{A}$ on its subalgebra $\mathcal{A}(2)$. These arise as the cohomology of $\text{v}_2$ self maps $\Sigma^7 Z \longrightarrow Z$, as in arXiv:1608.06250 [math.AT]. There are $256$ $\mathbb{F}_2$ points in this variety, arising from $16$ such actions of $Sq^8$ and, for each such, $16$ actions of $Sq^{16}$. We describe in similar fashion the 1600 $\mathcal{A}$ actions on $\mathcal{A}(2)$ found by Roth(1977) and the inclusion of the variety of symmetric actions into the variety of all actions. We also describe two related varieties of $\mathcal{A}$ actions, the maps between these and the behavior of Spanier-Whitehead duality on these varieties. Finally, we note that the actions which have been used in the literature correspond to the simplest choices, in which all the coordinates equal zero.